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In this chapter we consider LU factorizations of Hermitian and positive definite matrices. Recall that a matrix \({\boldsymbol {A}}\in {\mathbb {C}}^{n\times n}\) is Hermitian if A∗ = A, i.e., \(a_{ji}=\overline {a}_{ij}\) for all i, j. A real Hermitian matrix is symmetric. Since \(a_{ii}=\overline {a}_{ii}\) the diagonal elements of a Hermitian matrix must be real.

The factorization is easily verified. Here \( \mathit\mathbf{U}=\mathit\mathbf{V}=\frac{1}{\sqrt{2}} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right], \)which is clearly unitary. Since in addition the matrix in the middle is diagonal, it follows that this is both a spectral and a singular value decomposition.

In this chapter we present a fast method for solving Ax = b, where A is the Poisson matrix ( 10.8).

The QR algorithm is a method to find all eigenvalues and eigenvectors of a matrix. In this chapter we give a brief informal introduction to this important algorithm.

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

No, we have seen that 4n3/3 operations are required in order to bring a matrix to upper triangular form using Householder transformations.

International audience; Observations of the form D + X, where D is a matrix representing information, and X is a random matrix representing noise, can be grouped into a compund observation matrix, on the same information + noise form. There are many ways the observations can be stacked into such a matrix, for instance vertically, horizontally, or quadratically. An unbiased estimator for the spectrum of D can be formulated for each stacking scenario in the case of Gaussian noise. We compare these spectrum estimators for the different stacking scenarios, and show that all kinds of stacking actually decrease the variance of the corresponding spectrum estimators when compared to just taking an average of the observations, and find which stacking is optimal in this sense. When the number of observations grow, however, it is shown that the difference between the estimators is marginal. with only the cases of vertical and horizontal stackings having a higher variance asymptotically.